Theory and Applications of the Conformal Bootstrap

Abstract

Critical phenomena are ubiquitous in theoretical physics, in topics ranging from condensed matter to string theory. In most examples, the critical points of second order phase transitions can be described in terms of conformal field theories (CFTs), relativistic quantum field theories that are invariant under conformal transformations. However, there are still many open questions regarding the nature of critical phenomena and their associated CFTs, starting from the most basic question of classifying such theories. In this thesis, we present known and new ideas about the conformal bootstrap, an innovative method which has the potential to numerically classify all possible conformal field theories and consequently, all critical phenomena. While in the past years the conformal bootstrap has been very successful in describing scalar theories, as it has numerically determined the critical exponent of the infamous 3-dimensional Ising model, the method has not been extended to constrain theories using correlators of non-scalar operators. Here, we develop the theoretical tools through which one can apply the bootstrap to fermionic theories and thus find constrains for known CFTs with interacting fermions, such as the Gross-Neveu models or N = 1 supersymmetric models. In order to extend the bootstrap technique for theories with fermions, we analytically derive the conformal blocks for the four-point function of spin-1/2 external operators. Using these results, we perform a systematic numerical analysis to constrain relativistic CFTs with fermionic operators, that have a small number of relevant scalars in their spectrum. Besides bounding all CFTs with fermionic operators, the bootstrap also places severe constrains on the spectrum of the N = 1 super-Ising model, the interacting superconformal field theory (SCFT) with the simplest operator content. In the second part of this thesis, we continue constraining the operator spectrum of SCFTs, this time analyzing the four-point of chiral operators of theories with four Poincare supercharges. While for N = 1 SCFTs, the critical exponents of the system are not a priori known, for N = 2 SCFTs the scaling dimensions of some operators can be determined through a technique called supersymmetric localization. Using the bootstrap for N = 2 SCFTs with a global O(N) symmetry, we strongly constrain the scaling dimension of the lowest-lying scalar, whose properties are not determined by supersymmetry constrains.